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Properties

Label 19.1.239...368.1
Degree $19$
Signature $[1, 9]$
Discriminant $-2.397\times 10^{29}$
Root discriminant \(35.18\)
Ramified primes see page
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $S_{19}$ (as 19T8)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^19 - 3*x^18 + 2*x^17 + 4*x^16 - 8*x^15 + 14*x^13 - 9*x^12 - 16*x^11 + 21*x^10 + 8*x^9 - 25*x^8 + 2*x^7 + 17*x^6 - 4*x^5 - 9*x^4 + 2*x^3 + 3*x^2 - x - 1)
 
gp: K = bnfinit(y^19 - 3*y^18 + 2*y^17 + 4*y^16 - 8*y^15 + 14*y^13 - 9*y^12 - 16*y^11 + 21*y^10 + 8*y^9 - 25*y^8 + 2*y^7 + 17*y^6 - 4*y^5 - 9*y^4 + 2*y^3 + 3*y^2 - y - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^19 - 3*x^18 + 2*x^17 + 4*x^16 - 8*x^15 + 14*x^13 - 9*x^12 - 16*x^11 + 21*x^10 + 8*x^9 - 25*x^8 + 2*x^7 + 17*x^6 - 4*x^5 - 9*x^4 + 2*x^3 + 3*x^2 - x - 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^19 - 3*x^18 + 2*x^17 + 4*x^16 - 8*x^15 + 14*x^13 - 9*x^12 - 16*x^11 + 21*x^10 + 8*x^9 - 25*x^8 + 2*x^7 + 17*x^6 - 4*x^5 - 9*x^4 + 2*x^3 + 3*x^2 - x - 1)
 

\( x^{19} - 3 x^{18} + 2 x^{17} + 4 x^{16} - 8 x^{15} + 14 x^{13} - 9 x^{12} - 16 x^{11} + 21 x^{10} + \cdots - 1 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $19$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[1, 9]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-239740209347874920937820362368\) \(\medspace = -\,2^{7}\cdot 7\cdot 60259\cdot 4440286064038502416537\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(35.18\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $2^{2}7^{1/2}60259^{1/2}4440286064038502416537^{1/2}\approx 173111311497788.5$
Ramified primes:   \(2\), \(7\), \(60259\), \(4440286064038502416537\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  $\Q(\sqrt{-37459\!\cdots\!43162}$)
$\card{ \Aut(K/\Q) }$:  $1$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  Yes
Index:  $1$
Inessential primes:  None

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $9$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:   $a$, $a^{18}-4a^{17}+6a^{16}-2a^{15}-6a^{14}+6a^{13}+8a^{12}-17a^{11}+a^{10}+20a^{9}-12a^{8}-13a^{7}+15a^{6}+2a^{5}-6a^{4}-2a^{3}+3a^{2}-1$, $a^{18}-3a^{17}+3a^{16}+a^{15}-6a^{14}+4a^{13}+6a^{12}-9a^{11}-2a^{10}+12a^{9}-8a^{8}-4a^{7}+9a^{6}-6a^{5}-2a^{4}+6a^{3}-2a^{2}-3a$, $a^{18}-3a^{17}+2a^{16}+4a^{15}-8a^{14}+14a^{12}-9a^{11}-16a^{10}+21a^{9}+7a^{8}-23a^{7}+a^{6}+16a^{5}-a^{4}-10a^{3}-a^{2}+3a+1$, $a^{18}-2a^{17}-a^{16}+6a^{15}-4a^{14}-9a^{13}+16a^{12}+4a^{11}-28a^{10}+10a^{9}+30a^{8}-25a^{7}-19a^{6}+29a^{5}+3a^{4}-17a^{3}+5a^{2}+4a-2$, $a^{17}-3a^{16}+2a^{15}+3a^{14}-6a^{13}+a^{12}+9a^{11}-6a^{10}-11a^{9}+12a^{8}+3a^{7}-7a^{6}+2a^{5}-2a^{3}+2a^{2}-1$, $3a^{18}-13a^{17}+25a^{16}-15a^{15}-29a^{14}+55a^{13}-a^{12}-85a^{11}+62a^{10}+89a^{9}-146a^{8}-15a^{7}+149a^{6}-57a^{5}-80a^{4}+39a^{3}+42a^{2}-19a-19$, $4a^{18}-8a^{17}-3a^{16}+20a^{15}-14a^{14}-24a^{13}+44a^{12}+17a^{11}-76a^{10}+9a^{9}+82a^{8}-37a^{7}-63a^{6}+32a^{5}+39a^{4}-16a^{3}-21a^{2}-2a+2$, $a^{18}-4a^{16}+a^{15}+9a^{14}-11a^{13}-9a^{12}+32a^{11}-3a^{10}-50a^{9}+25a^{8}+48a^{7}-45a^{6}-33a^{5}+40a^{4}+15a^{3}-21a^{2}-9a+1$ Copy content Toggle raw display (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 21087865.9745 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{9}\cdot 21087865.9745 \cdot 1}{2\cdot\sqrt{239740209347874920937820362368}}\cr\approx \mathstrut & 0.657326298247 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^19 - 3*x^18 + 2*x^17 + 4*x^16 - 8*x^15 + 14*x^13 - 9*x^12 - 16*x^11 + 21*x^10 + 8*x^9 - 25*x^8 + 2*x^7 + 17*x^6 - 4*x^5 - 9*x^4 + 2*x^3 + 3*x^2 - x - 1)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^19 - 3*x^18 + 2*x^17 + 4*x^16 - 8*x^15 + 14*x^13 - 9*x^12 - 16*x^11 + 21*x^10 + 8*x^9 - 25*x^8 + 2*x^7 + 17*x^6 - 4*x^5 - 9*x^4 + 2*x^3 + 3*x^2 - x - 1, 1);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^19 - 3*x^18 + 2*x^17 + 4*x^16 - 8*x^15 + 14*x^13 - 9*x^12 - 16*x^11 + 21*x^10 + 8*x^9 - 25*x^8 + 2*x^7 + 17*x^6 - 4*x^5 - 9*x^4 + 2*x^3 + 3*x^2 - x - 1);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^19 - 3*x^18 + 2*x^17 + 4*x^16 - 8*x^15 + 14*x^13 - 9*x^12 - 16*x^11 + 21*x^10 + 8*x^9 - 25*x^8 + 2*x^7 + 17*x^6 - 4*x^5 - 9*x^4 + 2*x^3 + 3*x^2 - x - 1);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$S_{19}$ (as 19T8):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A non-solvable group of order 121645100408832000
The 490 conjugacy class representatives for $S_{19}$
Character table for $S_{19}$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.
sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.8.0.1}{8} }{,}\,{\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.5.0.1}{5} }$ $17{,}\,{\href{/padicField/5.2.0.1}{2} }$ R ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }$ ${\href{/padicField/13.13.0.1}{13} }{,}\,{\href{/padicField/13.6.0.1}{6} }$ ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ $16{,}\,{\href{/padicField/19.3.0.1}{3} }$ $16{,}\,{\href{/padicField/23.3.0.1}{3} }$ $19$ ${\href{/padicField/31.11.0.1}{11} }{,}\,{\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.13.0.1}{13} }{,}\,{\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ $16{,}\,{\href{/padicField/41.3.0.1}{3} }$ ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.5.0.1}{5} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ $16{,}\,{\href{/padicField/47.3.0.1}{3} }$ ${\href{/padicField/53.13.0.1}{13} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ $15{,}\,{\href{/padicField/59.4.0.1}{4} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
 
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display 2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
2.4.4.2$x^{4} + 4 x^{3} + 4 x^{2} + 12$$2$$2$$4$$C_4$$[2]^{2}$
2.4.0.1$x^{4} + x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.9.0.1$x^{9} + x^{4} + 1$$1$$9$$0$$C_9$$[\ ]^{9}$
\(7\) Copy content Toggle raw display 7.2.1.1$x^{2} + 21$$2$$1$$1$$C_2$$[\ ]_{2}$
7.4.0.1$x^{4} + 5 x^{2} + 4 x + 3$$1$$4$$0$$C_4$$[\ ]^{4}$
7.6.0.1$x^{6} + x^{4} + 5 x^{3} + 4 x^{2} + 6 x + 3$$1$$6$$0$$C_6$$[\ ]^{6}$
7.7.0.1$x^{7} + 6 x + 4$$1$$7$$0$$C_7$$[\ ]^{7}$
\(60259\) Copy content Toggle raw display $\Q_{60259}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{60259}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{60259}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $9$$1$$9$$0$$C_9$$[\ ]^{9}$
\(444\!\cdots\!537\) Copy content Toggle raw display $\Q_{44\!\cdots\!37}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{44\!\cdots\!37}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $15$$1$$15$$0$$C_{15}$$[\ ]^{15}$